1. Relation for collision frequency

For a gas in a closed chamber, the collision frequency is proportional to the average speed of molecules:

$Z \propto \bar v$

And from kinetic theory,

$\bar v \propto \sqrt{T}$

Hence,

$Z \propto \sqrt{T}$

So for two temperatures,

$\frac{Z_2}{Z_1} = \sqrt{\frac{T_2}{T_1}}$

2. Convert temperatures to kelvin

At $27^\circ\mathrm{C}$:

$T_1 = 27 + 273 = 300\,\mathrm{K}$

At $127^\circ\mathrm{C}$:

$T_2 = 127 + 273 = 400\,\mathrm{K}$

3. Compute the new collision frequency

Given initial collision frequency is $Z$ at $300\,\mathrm{K}$, let the new collision frequency be $Z_2$.

Then,

$\frac{Z_2}{Z} = \sqrt{\frac{400}{300}} = \sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}}$

Therefore,

$Z_2 = \frac{2}{\sqrt{3}} Z$

4. Match with options

This corresponds to:

Option B: $\dfrac{2}{\sqrt{3}}Z$